Powers (how 2^(2017/2)=sqrt2*2^1008 works)?

I have (1-i)^2017=?
I know that (cistheta)^n=cis(ntheta)
=> ... =>
=2^1008-i2^1008

The problem is HOW am I solving:
2^(2017/2)=sqrt2*2^1008

1 Answer
Nov 29, 2017

First remember that:

sqrt(a^3)=sqrt(axxa^2)=>asqrta
a^(x/y)=root[y] (a^x)
sqrt(a^x)=a^(x/2)

We know that 2^(2017/2) = sqrt(2^2017)

By our second and third rule, we know that sqrt(2^2017)=sqrt(2xx2^2016)=>2^(2016/2)sqrt2

When simplified, it becomes 2^1008sqrt2